High-precision position tracking AILC control method for industrial mechanical arm under complex working conditions

Abstract

The invention discloses a high-precision position tracking self-adaptive iterative learning control method for a mechanical arm under complex working conditions, which comprises the following steps of: firstly, determining a kinetic model of the mechanical arm, determining a position tracking error system of the mechanical arm, designing virtual control input and actual control input by utilizing a self-adaptive iterative learning control method on the basis, and finally, determining a position tracking error system of the mechanical arm; meanwhile, based on Taylor series-fuzzy approximation theory design, unmodeled dynamics is approximated. In addition, it is proved that the tracking error is convergent based on the Lyapunov stability theory, and meanwhile the effectiveness of the tracking error is verified through simulation. According to the invention, the finite time position tracking of the mechanical arm can be effectively controlled, the high-precision tracking control of the position of the mechanical arm is realized, and the product quality and the working efficiency can be improved.

Description

Technical Field

[0001] The invention belongs to the technical field of adaptive iterative learning control, and in particular relates to a Taylor series fuzzy AILC control method for an industrial robot arm under complex working conditions. Background Art

[0002] The robotic arm is an important part of intelligent manufacturing. The characteristics of automation, high precision and flexibility of the robotic arm make it an important means to realize intelligent manufacturing. The industrial robotic arm is a mechanical and electronic device with the functions of anthropomorphic arms, wrists and hands. It can accurately move and operate various objects or tools according to the preset spatial position and posture instructions that change over time to meet the diverse needs of industrial production. It has the characteristics of automation, high precision, strong flexibility, etc., and can work for a long time in harsh environments, greatly improving production efficiency and product quality.

[0003] High-speed and accurate trajectory tracking and positioning are the premise and guarantee for the robot to perform various complex tasks. In an industrial environment, the robot needs to move according to a predetermined trajectory to complete various complex tasks. Therefore, the tracking control problem of the robot is of great significance to improving production efficiency and product quality. In an industrial environment, the robot often needs to perform repetitive tasks, such as assembly and welding of parts on the assembly line. In order to ensure product quality and production efficiency, the repetitive tracking control of the robot is essential. It requires the robot to maintain a high degree of accuracy and consistency when executing the same trajectory multiple times, so as to meet production requirements.

[0004] In order for industrial robots to achieve high-precision tracking control within a limited time, these models need to be accurately described and analyzed. However, due to the influence of nonlinear factors such as friction and load changes, the accuracy of the model is often difficult to guarantee, which increases the difficulty of control. There are various uncertainties in the robot system, such as changes in parameters such as joint mass, length, friction coefficient, and external environmental interference. These uncertainties will affect the motion trajectory and accuracy of the robot. When repeatedly performing the same task, the robot may encounter some unmodeled interferences, such as changes in workpiece size, assembly errors, etc. These interferences will affect the repetitive tracking accuracy of the robot. Under high-speed movement, the flexible joint components of the robot will deform, thereby affecting the tracking accuracy of the end of the robot. Based on the above discussion, the development of high-precision position tracking control technology for industrial robots under complex working conditions is necessary.

[0005] Adaptive iterative learning control is an effective control method to solve the problem of repeated tracking control of uncertain nonlinear systems. It gradually approaches the ideal trajectory of the unknown system by iteratively adjusting the control algorithm. As the number of iterations increases, it quickly achieves the task requirements of accurate tracking. Adaptive iterative learning control technology can solve the problem of high-precision position tracking control of industrial robotic arms. Summary of the invention

[0006] The purpose of the present invention is to provide a high-precision position tracking adaptive iterative learning control method for a robotic arm under complex working conditions, to achieve tracking control of an industrial robotic arm in a finite time interval, to control the end of the robotic arm using an adaptive iterative learning control technology based on Taylor series-fuzzy approximation, to increase the accuracy of position control, to achieve precise motion control of the end of the robotic arm, and to improve the trajectory tracking accuracy and robustness of the robotic arm.

[0007] The technical solution adopted by the present invention is a high-precision position tracking adaptive iterative learning control method for a robot arm under complex working conditions, which is specifically implemented according to the following steps:

[0008] Step 1: Determine the dynamic model and control target of the robot arm;

[0009] Step 2: Establish an intelligent model for the robot model and determine the position tracking error system of the robot;

[0010] Step 3, design virtual control input and actual torque control input;

[0011] Step 4: Analyze the stability of the designed robotic arm control system.

[0012] The present invention is also characterized in that:

[0013] In step 1,

[0014] Assume y r (t) is the specified position trajectory, k represents the number of iterations, that is, the number of times the robot arm repeats the action;

[0015] The dynamic model of the robotic arm is as follows:

[0016]

[0017]

[0018] y k =x 1,k

[0019] The control objective is to design a high-precision position tracking adaptive iterative learning controller u for the robotic arm system described above. k , implemented on a finite time interval [0,T]:

[0020] lim k→∞ ‖y k (t)-y r (t)‖=0

[0021] That is, the robot system can move accurately in a limited time under the control of the controller, achieving high-precision position tracking control. 1,k , x 2,k ∈R and u k ∈R is the state variable displacement, velocity and control input joint torque of the manipulator system in nonlinear parameterized form, y k ∈R is the displacement output of the robot, is the unmodeled dynamic part with unknown time-varying parameters θ(t), including model uncertainty and unknown time-varying disturbances, where

[0022] In step 2,

[0023] Based on the approximation theory, a Taylor series fuzzy model is established for the uncertain part of the robot model, and the position tracking error system of the robot is determined:

[0024] z 1,k =y k (t)-y r (t).

[0025] Step 2 is as follows:

[0026] Step 2.1: Process the uncertain time-varying parameters in the robot model:

[0027] Expand the unknown time-varying parameters θ(t) in the model into a Taylor series:

[0028]

[0029] Where Φ(t)∈R m×1 is a power series term about time; M T ∈R m×1 is the corresponding weight matrix; δ θ (t) is the residual error after Fourier series expansion, which is bounded by

[0030] Step 2.2: Establish a new Taylor series expansion-fuzzy logic system approximator and model G(χ, t) and f(χ, θ(t)) as:

[0031] G(χ,t)=W T S(χ,M T Φ(t))

[0032] f(χ,θ(t))=W T S(χ,M T Φ(t)+δ θ (t))+δ f

[0033] =WT S(χ,M T Φ(t))+δ(χ,t)

[0034] in,

[0035] δ(χ,t)=δ f +W T S(χ,M T Φ(t)+δ θ (t))-W T S(χ,M T Φ(t))

[0036] S(χ,M T Φ(t)+δ θ (t)) is related to the state parameters χ and M T Φ(t)+δ θ (t) is the n-th order basis function vector; δ f is the approximation error; the weight matrices W and M are bounded, satisfying ‖W‖≤w m ,‖M‖≤m a , w m ,m a is an unknown normal bound, δ(χ,t) is bounded, and in, represents the upper bound of δ(χ,t), in and is an estimate of the unknown weight vectors W and M, and Represent the estimation errors between the estimated and actual values ​​of W and M respectively;

[0037] Step 2.3, unknown boundary processing and error function design:

[0038] The unknown upper bounds of each error term are handled using the following typical sequence of convergent series: And satisfy

[0039] At the beginning of each iteration, the initial error value of the end of the robot arm satisfies |z i,k (0)|=∈ i,k , i=1,...,n, where ∈ i,k is a convergent series sequence, design a new function z φ,k , solve the problem of initial state error and construct a new function z φ,k for:

[0040]

[0041] φ k (t)=∈k e -ηt

[0042] in, is a saturation function, defined as

[0043]

[0044] Among them, φ k (t) is a modified time-varying boundary layer.

[0045] Step 3 is as follows:

[0046] Step 3.1: Design the virtual control input α according to the first subsystem 1,k

[0047] According to the position tracking error and the Lyapunov function, the virtual control input is designed as: in is an estimate of the upper bound N1 of the unknown parameters, η1 is the gain value that needs to be designed;

[0048] Step 3.2: Design actual control torque input u k And parameter adaptation law The actual control torque is designed to be

[0049]

[0050] Select the parameter update law as

[0051]

[0052]

[0053] The new error function z 2φ,k for:

[0054]

[0055]

[0056]

[0057]

[0058] The Lyapunov function is shown below:

[0059]

[0060] Step 4 is as follows:

[0061] Because ‖z φ,k (0)‖2 =0≤‖z φ,k (T)‖ 2 , take the Lyapunov function as:

[0062]

[0063] have

[0064] Among them, z φ,k =[z 1φ,k ,z 2φ,k ] T ,

[0065] have to

[0066] definition Therefore, the above formula can be rewritten as

[0067]

[0068] According to the properties of convergent series, Therefore, V0(k) is bounded and has therefore k is arbitrary, for have to

[0069]

[0070] is bounded, through the concept of convergent series sequence, Δ k is bounded, and t∈[0,T], so we have is also bounded; for and For any k, we have is bounded, so It is bounded;

[0071] For any k, V 2,k (t) is bounded, so x 1,k ,x 2,k , and are all bounded, u k There are also boundaries, and is bounded, and by z iφ,k It is uniformly continuous, which proves that in a finite time interval, under complex and uncertain working conditions, the position tracking error of the robotic arm system can converge to zero as the number of iterations increases, that is, the task of high-precision tracking in a finite time interval is completed. BRIEF DESCRIPTION OF THE DRAWINGS

[0072] Figure 1 It is the overall flow chart of the Taylor series fuzzy AILC control method of the industrial robot arm under complex working conditions of the present invention. DETAILED DESCRIPTION

[0073] The present invention is described in detail below with reference to the accompanying drawings and specific embodiments.

[0074] As described in the background technology, high-precision control technology for industrial robot arms under complex working conditions has yet to be developed. The present invention combines approximation theory knowledge with an adaptive iterative learning control method to achieve high-precision position tracking control of the robot arm in a limited time interval, ensuring that the motion trajectory of the robot arm is consistent with the set motion curve, quickly responding to process requirements, meeting the requirements of high-precision processes, and improving production efficiency and product quality.

[0075] The invention discloses a high-precision position tracking adaptive iterative learning control method for a mechanical arm under complex working conditions. Figure 1 , follow the steps below:

[0076] Step 1: Determine the dynamic model and control target of the robot arm according to the kinematic law of the robot arm;

[0077] In step 1,

[0078] Assume y r (t) is the specified position trajectory, k represents the number of iterations, that is, the number of times the robot arm repeats the action;

[0079] The dynamic model of an industrial robot arm describes the relationship between the motion state of the robot arm and the torque it is subjected to. This relationship is particularly complex under complex working conditions because the robot arm may be affected by a variety of external factors. Considering complex working conditions (such as friction, temperature changes, and load changes), the robot arm model can be established as the following uncertain nonlinear parameterized system. The finite-time high-precision position tracking control problem of the robot arm can be regarded as a finite-time high-precision position tracking control problem of an uncertain nonlinear parameterized system. The dynamic model of the robot arm is as follows:

[0080]

[0081]

[0082] y k =x 1,k

[0083] The control objective is to design a high-precision position tracking adaptive iterative learning controller u for the robotic arm system described above. k , implemented on a finite time interval [0,T]:

[0084] limk→∞ ‖y k (t)-y r (t)‖=0

[0085] That is, the robot system can move accurately in a limited time under the control of the controller, achieving high-precision position tracking control. 1,k , x 2,k ∈R and u k ∈R is the state variable displacement, velocity and control input joint torque of the manipulator system in nonlinear parameterized form, y k ∈R is the displacement output of the robot, is the unmodeled dynamic part with unknown time-varying parameters θ(t), including model uncertainty and unknown time-varying disturbances, where

[0086] Step 2: Based on Taylor series-fuzzy approximation theory, an intelligent model is established for the robot model, and the position tracking error system of the robot is determined; in step 2,

[0087] Based on the approximation theory, a Taylor series fuzzy model is established for the uncertain part of the robot model, and the position tracking error system of the robot is determined:

[0088] z 1,k =y k (t)-y r (t).

[0089] Step 2 is as follows:

[0090] Step 2.1: Process the uncertain time-varying parameters in the robot model:

[0091] Expand the unknown time-varying parameters θ(t) in the model into a Taylor series:

[0092]

[0093] Where Φ(t)∈R m×1 is a power series term about time; M T ∈R m×1 is the corresponding weight matrix; δ θ (t) is the residual error after Fourier series expansion, which is bounded by

[0094] Step 2.2: Establish a new Taylor series expansion-fuzzy logic system approximator and model G(χ, t) and f(χ, θ(t)) as:

[0095] G(χ,t)=W T S(χ,M T Φ(t))

[0096] f(χ,θ(t))=W T S(χ,M T Φ(t)+δ θ (t))+δ f

[0097] =W T S(χ,M T Φ(t))+δ(χ,t)

[0098] in,

[0099] δ(χ,t)=δ f +W T S(χ,M T Φ(t)+δ θ (t))-W T S(χ,M T Φ(t))

[0100] S(χ,M T Φ(t)+δ θ (t)) is related to the state parameters χ and M T Φ(t)+δ θ (t) is the n-th order basis function vector; δ f is the approximation error; the weight matrices W and M are bounded, satisfying ‖W‖≤w m ,‖M‖≤m a , w m ,m a is an unknown normal bound, δ(χ,t) is bounded, and in, represents the upper bound of δ(χ,t), in and is an estimate of the unknown weight vectors W and M, and Represent the estimation errors between the estimated and actual values ​​of W and M respectively;

[0101] Step 2.3, unknown boundary processing and error function design:

[0102] The unknown upper bounds of each error term are handled using the following typical sequence of convergent series: k=1,2,…,l≥2, and satisfies

[0103] At the beginning of each iteration, the initial error value of the end of the robot arm satisfies |z i,k (0)|=∈ i,k , i=1,...,n, where ∈ i,k Design a new function z under the condition that it is a convergent series sequenceφ,k , solve the problem of initial state error and construct a new function z φ,k for:

[0104]

[0105] φ k (t)=∈ k e -ηt

[0106] in, is a saturation function, defined as

[0107]

[0108] Among them, φ k (t) is a modified time-varying boundary layer.

[0109] Step 3: Based on the adaptive iterative learning control theory, design the virtual control input and the actual torque control input;

[0110] Step 3 is as follows:

[0111] Step 3.1: Design the virtual control input α according to the first subsystem 1,k

[0112] According to the position tracking error and the Lyapunov function, the virtual control input is designed as: in is an estimate of the upper bound N1 of the unknown parameters, η1 is the gain value that needs to be designed;

[0113] Step 3.2: Design actual control torque input u k And parameter adaptation law

[0114] The actual control torque is designed to be

[0115]

[0116] Select the parameter update law as

[0117]

[0118]

[0119] The new error function z 2φ,k for:

[0120]

[0121]

[0122]

[0123]

[0124] The Lyapunov function is shown below:

[0125]

[0126] Step 4: Based on the Lyapunov stability theorem, analyze the stability of the designed robotic arm control system.

[0127] Step 4 is as follows:

[0128] When t = 0, for any number of iterations k, we have In this case, for the uncertain robotic arm control system, combined with the Taylor series-fuzzy approximation theory, the actual high-precision tracking controller (the actual control torque given in step 3.2) can ensure that even if there is an error in the initial position in a finite time interval, the position tracking error of the robotic arm system can converge to zero as the number of iterations increases, thus completing the task of high-precision complete tracking.

[0129] Because ‖z φ,k (0)‖ 2 =0≤‖z φ,k (T)‖ 2 , take the Lyapunov function as:

[0130]

[0131] have

[0132] Among them, z φ,k =[z 1φ,k ,z 2φ,k ] T ,

[0133] have to

[0134] definition Therefore, the above formula can be rewritten as

[0135]

[0136] According to the properties of convergent series, Therefore, V0(k) is bounded and has therefore k is arbitrary, for have to

[0137]

[0138] is bounded, through the concept of convergent series sequence, Δ k is bounded, and t∈[0,T], so we have is also bounded; for and For any k, we have is bounded, so It is bounded;

[0139] For any k, V 2,k (t) is bounded, so x 1,k ,x 2,k , and are all bounded, u k There are also boundaries, and is bounded, and by z iφ,k It is uniformly continuous, which proves that in a finite time interval, under complex and uncertain working conditions, the position tracking error of the robotic arm system can converge to zero as the number of iterations increases, that is, the task of high-precision tracking in a finite time interval is completed.

[0140] Example 1

[0141] The invention discloses a high-precision position tracking adaptive iterative learning control method for a mechanical arm under complex working conditions. Figure 1 , follow the steps below:

[0142] Step 1: Determine the dynamic model and control target of the robot arm according to the kinematic law of the robot arm;

[0143] Step 2: Based on Taylor series-fuzzy approximation theory, an intelligent model is established for the robot model, and the position tracking error system of the robot is determined;

[0144] Step 3: Based on the adaptive iterative learning control theory, design the virtual control input and the actual torque control input;

[0145] Step 4: Based on the Lyapunov stability theorem, analyze the stability of the designed robotic arm control system.

[0146] Example 2

[0147] Under complex working conditions, the dynamic model of the industrial robot arm is the key to describing its motion characteristics. By establishing accurate dynamic models and equations, and considering the influence of various complex factors, accurate control and optimized performance of the robot arm can be achieved. However, in practice, factors such as the structure, mass distribution, and joint characteristics of the robot arm are considered, and various complex factors such as external loads, environmental changes, and possible interference are also incorporated. The accuracy of the model is difficult to guarantee, so the end control of the robot arm becomes more difficult. Considering these factors, the dynamic model of the robot arm can be modeled as a nonlinear parameterized system with time-varying parameters and unmodeled dynamics. The present invention only provides theoretical guidance, mainly considering the precise position tracking control of the finite time interval of the uncertain nonlinear parameterized robot arm system.

[0148] The control goal of the present invention is to design a high-precision position tracking adaptive iterative learning controller for industrial robot arm systems under complex working conditions. k (t), implemented on a finite time interval [0,T]:

[0149] lim k→∞ ‖y k (t)-y r (t)‖=0

[0150] That is, the industrial robot arm system can move accurately in a limited time under the action of the controller to achieve high-precision position tracking control. In the present invention, it is assumed that y r (t) is the specified position trajectory. k ∈R represents the output displacement of the robot arm, and k represents the number of iterations, which here represents the number of times the robot arm repeats the action.

[0151] In order to achieve the above object, the present invention adopts the following technical solution, which specifically includes the following steps:

[0152] Step 1: Build a dynamic model of the industrial robot arm.

[0153] The dynamic model of an industrial robot arm describes the relationship between the motion state of the robot arm and the torque it is subjected to. This relationship is particularly complex under complex working conditions because the robot arm may be affected by a variety of external factors. Considering complex working conditions (such as friction, temperature changes, and load changes), the robot arm model can be established as the following uncertain nonlinear parameterized system. The finite-time high-precision position tracking control problem of the robot arm can be regarded as a finite-time high-precision position tracking control problem of an uncertain nonlinear parameterized system. The dynamic model of the robot arm is expressed as follows:

[0154]

[0155]

[0156] yk =x 1,k

[0157] Among them, x 1,k , x 2,k ∈R and u k ∈R is the state variable displacement, velocity and control input joint torque of the manipulator system in nonlinear parameterized form, y k ∈R is the displacement output of the robot arm. is the unmodeled dynamic part with unknown time-varying parameters θ(t), including model uncertainty and unknown time-varying disturbances.

[0158] Step 2: Based on the approximation theory, a Taylor series fuzzy model is established for the uncertain part of the robot model, and the position tracking error system of the robot is determined:

[0159] Step 3: Based on the adaptive iterative learning control theory, a finite time high-precision position tracking control method for the robotic arm system is given. The specific implementation steps are as follows:

[0160] Step 3.1: Design the virtual control input α according to the first subsystem 1,k

[0161] According to the position tracking error and selecting the appropriate Lyapunov function, the virtual control input is designed as: in is an estimate of the upper bound N1 of the unknown parameters, η1 is the gain value that needs to be designed.

[0162] Step 3.2: Design actual control torque input u k And parameter adaptation law

[0163] The actual control torque is designed to be

[0164]

[0165] Select the parameter update law as

[0166]

[0167]

[0168]

[0169] Among them, the new error function z 2φ,k for:

[0170]

[0171]

[0172]

[0173]

[0174] Step 4: Based on the above results, the following conclusions are drawn for high-precision position tracking of the robot arm. When t = 0, for any number of iterations k, In this case, for the uncertain robotic arm control system, combined with the Taylor series-fuzzy approximation theory, the actual high-precision tracking controller (the actual control torque given in step 3.2) can ensure that even if there is an error in the initial position in a finite time interval, the position tracking error of the robotic arm system can converge to zero as the number of iterations increases, thus completing the task of high-precision complete tracking.

[0175] Example 3

[0176] The high-precision position tracking adaptive iterative learning control method of the mechanical arm under complex working conditions of the present invention is specifically implemented as follows:

[0177] Step 1: Build a dynamic model of the industrial robot arm.

[0178] Considering complex working conditions such as friction, temperature change, and load change, the robot model can be established as the following uncertain nonlinear parameterized system. The robot dynamics model is expressed as follows:

[0179]

[0180]

[0181] y k =x 1,k

[0182] Among them, x 1,k , x 2,k ∈R and u k ∈R is the state variable displacement, velocity and control input joint torque of the manipulator system in nonlinear parameterized form, y k ∈R is the displacement output of the robot arm. is the unmodeled dynamic part with unknown time-varying parameters θ(t), including model uncertainty and unknown time-varying disturbances.

[0183] Step 2: Based on the new approximation theory, a Taylor series fuzzy model is established for the uncertain part of the robot model, and the position tracking error system of the robot is determined:

[0184] z 1,k=y k (t)-y r (t)

[0185] The present invention considers the position tracking control of the robot arm under complex working conditions with uncertain robot arm model and unknown interference and with initial position error in each iteration, and only provides theoretical reference.

[0186] Step 2.1 Process the uncertain time-varying parameters in the robot model through Taylor series expansion:

[0187] Expand the unknown time-varying parameters θ(t) in the model into a Taylor series

[0188]

[0189] where Φ(t)∈R m×1 is a power series term about time; M T ∈R m×1 is the corresponding weight matrix; δ θ (t) is the residual error after Fourier series expansion, which is bounded by

[0190] Step 2.2 Establish a new Taylor series expansion-fuzzy logic system approximator and model G(χ,t) and f(χ,θ(t)) as

[0191] G(χ,t)=W T S(χ,M T Φ(t))

[0192] f(χ,θ(t))=W T S(χ,M T Φ(t)+δ θ (t))+δ f

[0193] =W T S(χ,M T Φ(t))+δ(χ,t)

[0194] in,

[0195] δ(χ,t)=δ f +W T S(χ,M T Φ(t)+δ θ (t))-W T S(χ,M T Φ(t))

[0196] S(χ,M T Φ(t)+δ θ (t)) is related to the state parameters χ and M T Φ(t)+δθ (t) is the n-th order basis function vector; δ f is the approximation error; the fuzzy weight matrices W and M are bounded, satisfying ‖W‖≤w m ,‖M‖≤m a , w m ,m a is an unknown normal bound. δ(χ,t) is bounded, and has in, represents the upper bound of δ(χ,t). in and is an estimate of the unknown weight vectors W and M, and Represent the estimation errors between the estimated and actual values ​​of W and M, respectively.

[0197] The approximation error of the approximator is

[0198]

[0199] In the above formula,

[0200]

[0201] in, The boundary of the remainder d is

[0202] Step 2.3 Design of new error function and treatment of unknown upper bound:

[0203] For the treatment of unknown upper bounds of each error term, the following typical convergent series sequence is introduced:

[0204] satisfy

[0205] At the beginning of each iteration, the initial error value should satisfy |z i,k (0)|=∈ i,k , i=1,…,n, where i,k is a convergent series sequence;

[0206] Construct a new function z φ,k for

[0207]

[0208] φ k (t)=∈ k e -ηt

[0209] in, is a saturation function, expressed as

[0210]

[0211] Among them, φ k (t) is a modified time-varying layer boundary.

[0212] Step 3: Based on the adaptive iterative learning control theory, a finite time high-precision position tracking control method for the robotic arm system is given. The specific implementation steps are as follows:

[0213] Step 4: Stability analysis of the robot tracking control system

[0214] Example 4

[0215] The high-precision position tracking adaptive iterative learning control method of the mechanical arm under complex working conditions of the present invention is specifically implemented as follows:

[0216] Step 1: Build a dynamic model of the industrial robot arm.

[0217] Considering complex working conditions such as friction, temperature change, and load change, the robot model can be established as the following uncertain nonlinear parameterized system. The robot dynamics model is expressed as follows:

[0218]

[0219]

[0220] y k =x 1,k

[0221] Among them, x 1,k , x 2,k ∈R and u k ∈R is the state variable displacement, velocity and control input joint torque of the manipulator system in nonlinear parameterized form, y k ∈R is the displacement output of the robot arm. is the unmodeled dynamic part with unknown time-varying parameters θ(t), including model uncertainty and unknown time-varying disturbances.

[0222] Step 2: Based on the new approximation theory, a Taylor series fuzzy model is established for the uncertain part of the robot model, and the position tracking error system of the robot is determined.

[0223] Step 3: Based on the adaptive iterative learning control theory, a finite-time high-precision position tracking control method for the robotic arm system is given.

[0224] Step 4: Stability analysis of the robot tracking control system

[0225] Example 5

[0226] The high-precision position tracking adaptive iterative learning control method of the mechanical arm under complex working conditions of the present invention is specifically implemented as follows:

[0227] Step 1: Build a dynamic model of the industrial robot arm.

[0228] Step 2: Based on the new approximation theory, a Taylor series fuzzy model is established for the uncertain part of the robot model, and the position tracking error system of the robot is determined:

[0229] z 1,k =y k (t)-y r (t)

[0230] The present invention considers the position tracking control of the robot arm under complex working conditions with uncertain robot arm model and unknown interference and with initial position error in each iteration, and only provides theoretical reference.

[0231] Step 3: Based on the adaptive iterative learning control theory, a finite time high-precision position tracking control method for the robotic arm system is given. The specific implementation steps are as follows:

[0232] Step 3.1: Design the virtual control input α according to the first subsystem 1,k

[0233] Select a new error function z 1φ,k and z 2φ,k as follows:

[0234]

[0235]

[0236]

[0237]

[0238] in, The first subsystem for the dynamics model of the robotic arm system:

[0239]

[0240] According to the position tracking error, the Lyapunov function is selected as In order to make Negative semi-definite, the virtual controller is designed as follows:

[0241]

[0242] in is the estimate of the unknown parameter N1, η1 is the gain value to be designed, and the parameter adaptation law is Step 3.2: Design actual control torque input u k And parameter adaptation law Select a new error function z 2φ,k for:

[0243]

[0244] Select the Lyapunov function

[0245]

[0246] Taking its derivative, we get

[0247]

[0248] The actual control torque is designed to be

[0249]

[0250] Select the parameter update law as

[0251]

[0252]

[0253]

[0254] in,

[0255]

[0256]

[0257] Step 4: Stability analysis of the robot tracking control system

[0258] Example 6

[0259] The high-precision position tracking adaptive iterative learning control method of the mechanical arm under complex working conditions of the present invention is specifically implemented as follows:

[0260] Step 1: Build a dynamic model of the industrial robot arm.

[0261] Considering complex working conditions such as friction, temperature change, and load change, the robot model can be established as the following uncertain nonlinear parameterized system. The robot dynamics model is expressed as follows:

[0262]

[0263]

[0264] y k =x 1,k

[0265] Among them, x 1,k , x 2,k ∈R and u k ∈R is the state variable displacement, velocity and control input joint torque of the manipulator system in nonlinear parameterized form, y k ∈R is the displacement output of the robot arm. is the unmodeled dynamic part with unknown time-varying parameters θ(t), including model uncertainty and unknown time-varying disturbances.

[0266] Step 2: Based on the new approximation theory, a Taylor series fuzzy model is established for the uncertain part of the robot model, and the position tracking error system of the robot is determined:

[0267] z 1,k =y k (t)-y r (t)

[0268] The present invention considers the position tracking control of the robot arm under complex working conditions with uncertain robot arm model and unknown interference and with initial position error in each iteration, and only provides theoretical reference.

[0269] Step 2.1 Process the uncertain time-varying parameters in the robot model through Taylor series expansion:

[0270] Expand the unknown time-varying parameters θ(t) in the model into a Taylor series

[0271]

[0272] where Φ(t)∈R m×1 is a power series term about time; M T ∈R m×1 is the corresponding weight matrix; δ θ (t) is the residual error after Fourier series expansion, which is bounded by

[0273] Step 2.2 Establish a new Taylor series expansion-fuzzy logic system approximator and model G(χ,t) and f(χ,θ(t)) as

[0274] G(χ,t)=W T S(χ,M T Φ(t))

[0275] f(χ,θ(t))=W T S(χ,M TΦ(t)+δ θ (t))+δ f

[0276] =W T S(χ,M T Φ(t))+δ(χ,t)

[0277] in,

[0278] δ(χ,t)=δ f +W T S(χ,M T Φ(t)+δ θ (t))-W T S(χ,M T Φ(t))

[0279] S(χ,M T Φ(t)+δ θ (t)) is related to the state parameters χ and M T Φ(t)+δ θ (t) is the n-th order basis function vector; δ f is the approximation error; the fuzzy weight matrices W and M are bounded, satisfying ‖W‖≤w m ,‖M‖≤m a , w m ,m a is an unknown normal bound. δ(χ,t) is bounded, and has in, represents the upper bound of δ(χ,t). in and is an estimate of the unknown weight vectors W and M, and Represent the estimation errors between the estimated and actual values ​​of W and M, respectively.

[0280] The approximation error of the approximator is

[0281]

[0282] In the above formula,

[0283]

[0284] in, The boundary of the remainder d is

[0285] Step 2.3 Design of new error function and treatment of unknown upper bound:

[0286] For the treatment of unknown upper bounds of each error term, the following typical convergent series sequence is introduced:

[0287] satisfy

[0288] At the beginning of each iteration, the initial error value should satisfy |z i,k (0)|=∈ i,k , i=1,...,n, where i,k is a convergent series sequence;

[0289] Construct a new function z φ,k for

[0290]

[0291] φ k (t)=∈ k e -ηt

[0292] in, is a saturation function, expressed as

[0293]

[0294] Among them, φ k (t) is a modified time-varying layer boundary.

[0295] Step 3: Based on the adaptive iterative learning control theory, a finite time high-precision position tracking control method for the robotic arm system is given. The specific implementation steps are as follows:

[0296] Step 3.1: Design the virtual control input α according to the first subsystem 1,k

[0297] Select a new error function z 1φ,k and z 2φ,k as follows:

[0298]

[0299]

[0300]

[0301]

[0302] in,

[0303] The first subsystem for the dynamics model of the robotic arm system:

[0304]

[0305] According to the position tracking error, the Lyapunov function is selected as

[0306] In order to make Negative semi-definite, the virtual controller is designed as follows:

[0307]

[0308] in is the estimate of the unknown parameter N1, η1 is the gain value to be designed,

[0309] The parameter adaptation law is

[0310] Step 3.2: Design actual control torque input u k And parameter adaptation law Select a new error function z 2φ,k for:

[0311]

[0312] Select the Lyapunov function

[0313]

[0314] Taking its derivative, we get

[0315]

[0316] The actual control torque is designed to be

[0317]

[0318] Select the parameter update law as

[0319]

[0320]

[0321]

[0322] in,

[0323]

[0324]

[0325] Step 4: Stability analysis of the robot tracking control system This part is a detailed proof of the conclusion. The specific process is as follows:

[0326] Because ‖z φ,k (0)‖ 2 =0≤‖z φ,k (T)‖2 , take the Lyapunov function as:

[0327]

[0328] have Among them, z φ,k =[z 1φ,k ,z 2φ,k ] T , Available

[0329]

[0330] definition Therefore, the above formula can be rewritten as

[0331]

[0332] According to the properties of convergent series, we know that Therefore, V0(k) is bounded and has therefore k is arbitrary, for Available

[0333]

[0334] is bounded. Through the concept of convergent series sequence, Δ k is bounded, and t∈[0,T], so we have is also bounded. and For any k, we have is bounded, so we can get is bounded. Most importantly, for any k, V 2,k (t) is bounded, so we have x 1,k ,x 2,k , and are all bounded. k is also bounded. is bounded, and by z iφ,k It is uniformly continuous, so it can be proved that in a finite time interval, under complex and uncertain working conditions, the position tracking error of the robotic arm system can converge to zero as the number of iterations increases, that is, the task of high-precision tracking in a finite time interval is completed.

Claims

1. A high-precision position tracking adaptive iterative learning control method for a robotic arm under complex working conditions, characterized in that: Follow the steps below to implement it: Step 1: Determine the dynamic model and control target of the robot arm; Step 2: Establish an intelligent model for the robot model and determine the position tracking error system of the robot; Step 3, design virtual control input and actual torque control input; Step 4: Analyze the stability of the designed robotic arm control system.

2. The high-precision position tracking adaptive iterative learning control method for a manipulator under complex working conditions according to claim 1 is characterized in that: In the step 1, Assume y r (t) is the specified position trajectory, y k ∈R represents the output displacement of the robot system, and k represents the number of iterations, that is, the number of times the robot repeats the action; The dynamic model of the robotic arm is as follows: The control objective is to design a high-precision position tracking adaptive iterative learning controller u for the robotic arm system described above. k , implemented on a finite time interval [0,T]: lim k→∞ ‖y k (t)-y r (t)‖=0 That is, the robot system can move accurately in a limited time under the action of the controller to achieve high-precision position tracking control, where x 1,k , x 2,k ∈R and u k ∈R is the state variable displacement, velocity and control input joint torque of the manipulator system in nonlinear parameterized form, y k ∈R is the displacement output of the robot, is the unmodeled dynamic part with unknown time-varying parameters θ(t), including model uncertainty and unknown time-varying disturbances, where 3. The high-precision position tracking adaptive iterative learning control method for a manipulator under complex working conditions according to claim 1 is characterized in that: In the step 2, Based on the approximation theory, a Taylor series fuzzy model is established for the uncertain part of the robot model, and the position tracking error system of the robot is determined: z 1,k =y k (t)-y r (t)。 4. The high-precision position tracking adaptive iterative learning control method for a robot arm under complex working conditions according to claim 3 is characterized in that: The step 2 is specifically as follows: Step 2.1: Process the uncertain time-varying parameters in the robot model: Expand the unknown time-varying parameters θ(t) in the model into a Taylor series: Where Φ(t)∈R m×1 is a power series term about time; M T ∈R m×1 is the corresponding weight matrix; δ θ (t) is the residual error after Fourier series expansion, which is bounded by Step 2.2: Establish a new Taylor series expansion-fuzzy logic system approximator and model G(χ, t) and f(χ, θ(t)) as: G(x,t)=W T S(x,M T Φ(t)) f(x,θ(t))=W T S(x,M T Φ(t)+δ θ (t))+δ f =W T S(x,M T Φ(t))+δ(x,t) in, δ(x,t)=δ f +W T S(x,M T Φ(t)+δ θ (t))-W T S(x,M T Φ(t)) S(χ,M T Φ(t)+δ θ (t)) is related to the state parameters χ and M T Φ(t)+δ θ (t) is the n-th order basis function vector; δ f is the approximation error; the weight matrices W and M are bounded, satisfying ‖W‖≤w m ,‖M‖≤m a , w m ,m a is an unknown normal bound, δ(χ,t) is bounded, and in, represents the upper bound of δ(χ,t), in and is an estimate of the unknown weight vectors W and M, and Represent the estimation errors between the estimated and actual values ​​of W and M respectively; Step 2.3, unknown boundary processing and error function design: The unknown upper bounds of each error term are handled using the following typical sequence of convergent series: k=1,2,…,l≥2, and satisfies At the beginning of each iteration, the initial error value of the robot end satisfies |z i,k (0)|=∈ i,k , i=1,...,n, where ∈ i,k Design a new function z under the condition that it is a convergent series sequence φ,k , solve the problem of initial state error and construct a new function z φ,k for: f k (t)=∈ k e -ηt in, is a saturation function, defined as Among them, φ k (t) is a modified time-varying boundary layer.

5. The high-precision position tracking adaptive iterative learning control method for a robot arm under complex working conditions according to claim 4, characterized in that: The step 3 is as follows: Step 3.1: Design the virtual control input α according to the first subsystem 1,k According to the position tracking error and the Lyapunov function, the virtual control input is designed as: in is an estimate of the upper bound N1 of the unknown parameters, η1 is the gain value that needs to be designed; Step 3.2: Design actual control torque input u k And parameter adaptation law The actual control torque is designed to be Select the parameter update law as 6. The high-precision position tracking adaptive iterative learning control method for a robot arm under complex working conditions according to claim 5, characterized in that: The new error function z 2φ,k for:

7. The high-precision position tracking adaptive iterative learning control method for a robot arm under complex working conditions according to claim 6, characterized in that: The Lyapunov function is shown in the following formula:

8. The high-precision position tracking adaptive iterative learning control method for a robot arm under complex working conditions according to claim 7, characterized in that: The step 4 is specifically as follows: Because ‖z φ,k (0)‖ 2 =0≤‖z φ,k (T)‖ 2 , take the Lyapunov function as: have where z φ,k = [z 1φ,k , z 2φ,k T , ​ have to definition Therefore, the above formula can be rewritten as According to the properties of convergent series, Therefore, V0(k) is bounded and has therefore k is arbitrary, for have to is bounded, through the concept of convergent series sequence, Δ k is bounded, and t∈[0,T], so we have is also bounded; for and For any k, we have is bounded, so It is bounded; For any k, V 2,k (t) is bounded, so x 1,k ,x 2,k , and are all bounded, u k There are also boundaries, and is bounded, and by z iφ,k It is uniformly continuous, which proves that in a finite time interval, under complex and uncertain working conditions, the position tracking error of the robotic arm system can converge to zero as the number of iterations increases, that is, the task of high-precision tracking in a finite time interval is completed.

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